Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

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The geometry of moduli spaces of sheaves

This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to repre- sentation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS 527 that the block of the Bernstein-Gelfand-Gelfand category O that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain cate- gories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail address: sasha@math.mit.edu Department of Mathematics, The University of Chicago, Chicago, Illinois 60637 E-mail address: ginzburg@math.uchicago.edu Max-Planck-Institut fur Mathematik, Gottfried-Claren-Strase 26, D-53 Bonn 3, Germany Current address: Mathematisches Institut, Universitat Freiburg, Albertstrase 23b, D-79104 Freiburg, Germany E-mail address: soergel@sun1.mathematik.uni-freiburg.de License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

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Koszul Duality Patterns in Representation Theory

— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a "Morita theorem" (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories.

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Deriving DG categories

Groupes et algébres de lie

/pdf/kategorie-perverse-garben-und-moduln-uber-den-koinvarianten-2kqypsu89y.pdf

Kategorie , perverse Garben und Moduln über den Koinvarianten zur Weylgruppe

Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces

The combinatorics of Harish-Chandra bimodules.

This article gives a self-contained treatment of the theory of Kazhdan-Lusztig polynomials with special emphasis on affine reflection groups. There are only a few new results but several new proofs. We close with a conjectural character formula for tilting modules, which formed the starting point of these investigations.

Wolfgang Soergel

Papers

Koszul Duality Patterns in Representation Theory

Kategorie , perverse Garben und Moduln über den Koinvarianten zur Weylgruppe

Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces

The combinatorics of Harish-Chandra bimodules.

Kazhdan-Lusztig polynomials and a combinatoric for tilting modules